Package diffeq. February 19, 2015

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1 Version Package diffeq February 19, 2015 Title Functions from the book Solving Differential Equations in R Author Karline Soetaert <karline.soetaert@nioz.nl> Maintainer Karline Soetaert <karline.soetaert@nioz.nl> Depends R (>= 2.01), desolve, rootsolve, bvpsolve, ReacTran, detestset Imports shape Suggests scatterplot3d Description Functions and examples from the book Solving Differential Equations in R by Karline Soetaert, Jeff R Cash and Francesca Mazzia. Springer, License GPL LazyData yes Repository CRAN Repository/R-Forge/Project bvpsolve Repository/R-Forge/Revision 223 Repository/R-Forge/DateTimeStamp :33:59 Date/Publication :40:11 NeedsCompilation no R topics documented: diffeq-package Coefficients rkmethodplot stability.multistep Index 10 1

2 2 diffeq-package diffeq-package Functions to create the figures of the book "solving differential equations in R" by Karline Soetaert, Jeff R. Cash and Francesca Mazzia. Published by Springer Description Details R package diffeq contains the functions for generating figures relating to differential equations Package: diffeq Type: Package Version: 1.0 License: GNU Public License 2 or above Author(s) See Also Karline Soetaert (Maintainer), Jeff Cash, Francesca Mazzia rkmethodplot for plotting the steps of runge-kutta methods stability.multistep, stability.bruteforce for plotting stability regions Coefficients for the BDF, AdamsMoulton, AdamsBashford coefficients. rkmethod from package desolve for the Runge-Kutta coefficients Examples ## Not run: ## show examples (see respective help pages for details) example(rkmethodplot) example(stability.multistep) example(bdf) ## open the directory with R sourcecode examples browseurl(paste(system.file(package = "diffeq"), "/doc/examples", sep = "")) ## show package vignette with how to use the package ## + source code of the vignette vignette("diffeq")

3 Coefficients 3 edit(vignette("diffeq")) ## End(Not run) Coefficients The coefficients of multistep methods Description Usage Coefficients alpha and beta of the Adams-Bashforth, Adams-Moulton and Backward differentiation formulae. sum_j=0^k alpha_j y_n-j = h sum_(j=0)^k beta_j f(x_n-j,y_n-j) For the BDF methods, the angle of the stability-region (the alpha of the A(alpha) stability, in radians is also given. BDF AdamsMoulton AdamsBashforth Author(s) See Also Karline Soetaert stability.multistep for plotting stability regions Examples ## Stability properties BDF stability.multistep(alpha = BDF$alpha[3,], beta = BDF$beta[3,], xlim = c(-7,7), ylim = c(-7,7)) stability.multistep(alpha = AdamsMoulton$alpha[3,], beta = AdamsMoulton$beta[3,], xlim = c(-7,7), ylim = c(-7,7) ) stability.multistep(alpha = AdamsBashforth$alpha[3,], beta = AdamsBashforth$beta[3,] ) ## Running a BDF

4 4 rkmethodplot # test model ode1 <- function (t, y) return(cos(t)*y ) h < times <- seq(from = 0, to = 20, by = h) yout <- vector (length = length(times)) yout[1] <- 1 # 3rd order BDF Alpha <- BDF$alpha [3,2:4] Beta <- BDF$beta[3,] bdf <- function(y, t, h, f, ys) { rootfun <- function(ynext) - ynext - sum(alpha * ys) + Beta * h * f(t + h, ynext) y <- multiroot(f=rootfun, start=y)$root ys[2:3] <- ys[1:2] ys[1] <- y } list (y = y, ys=ys) # Spinup uses Euler... Euler <- function(y, t, h, f) { fn <- f(t, y) ynext <- y + h * fn } list (y = ynext, fn = fn) for (i in 2:3) yout[i] <- Euler(yout[i-1], times[i-1], h, ode1)$y ys <- rev(yout[1:3]) # BDF steps for (i in 4:length(times)){ step <- bdf (y=yout[i-1], t=times[i-1], h=h, f=ode1, ys=ys) yout[i] <- step$y ys <- step$ys } rkmethodplot Plots the steps in runge-kutta methods Description...

5 rkmethodplot 5 Usage rkmethodplot (rk,...) Arguments Value rk A list containing the runge-kutta coefficients, implicit or explicit, e.g. matrix A, vectors b1, b2, codec. The list can be of type rkmethod, as defined in package desolve.... arguments passed to the plotting function. Returns nothing Author(s) See Also Karline Soetaert stability.bruteforce for plotting stability regions of Runge-Kutta methods. Examples # This to plot all runge kutta methods #RKS <- rkmethod() #for (i in 4:21) rkmethodplot( rkmethod(rks[i])) ## ## Figures A and B: Cash-Karp and Radau 5 steps ## par(mfrow=c(2,2)) rkmethodplot( rkmethod("rk45ck"), main="cash-karp") writelabel("a") rkmethodplot( rkmethod("irk5"), main="radau5") writelabel("b") rkmethodplot( rkmethod("rk45dp6"), main="dopri") writelabel("c") rkmethodplot( rkmethod("irk6l"), main="lobatto") writelabel("d") legend("bottomright", pch = c(16, 16, 1, NA), pt.cex = c(1.5, 1.5, 1), legend = c(expression(y[0]), expression(y[1]), "intermediary", "k"), col = c("grey", "black", "black", "black"), lty = c(na, NA, NA, 1), lwd = c(1, 1, 1, 2))

6 6 stability.multistep stability.multistep Plots the stability function of multistep methods Description... Usage stability.multistep (alpha, beta, add = FALSE, fill = NULL,...) stability.bruteforce (Rez = seq(-2, 2, by = 0.02), Imz = seq(-2, 2, by = 0.02), func = function (z) return(abs(1 + h*z) <=1), fill = "grey", cex = 1.5, add = FALSE,...) Arguments alpha beta add fill Rez Imz func cex alpha coefficients of the multistep method. beta coefficients of the multistep method. if TRUE, the new region will be added to the existing plot color of region to be filled The range in the real plane for testing stability The range in the imaginary plane for testing stability The function to be tested; default is test for the euler method. The relative size of the plotting symbol. If too small, the region will not be completely covered. If too large, it will extend beyond its boundaries.... arguments passed to the plotting function. Value A matrix with the boundary value. Author(s) Karline Soetaert See Also rkmethodplot for plotting runge-kutta method steps.

7 stability.multistep 7 Examples par(mfrow=c(2,2)) ## Stability regions for multistep methods # Adams-Bashforth stability.multistep(alpha = AdamsBashforth$alpha[2,], beta = AdamsBashforth$beta[2,], xlim = c(-3,1), ylim = c(-1.5, 1.5), fill = "black", main = "Adams-Bashforth") stability.multistep(alpha = AdamsBashforth$alpha[3,], beta = AdamsBashforth$beta[3,], add = TRUE, lty = 1, fill = "darkgrey") stability.multistep(alpha = AdamsBashforth$alpha[4,], beta = AdamsBashforth$beta[4,], add = TRUE, fill = "lightgrey", lty = 1) stability.multistep(alpha = AdamsBashforth$alpha[5,], beta = AdamsBashforth$beta[5,], add = TRUE, fill = "white", lty = 1) legend("topleft", fill = c("black", "darkgrey", "lightgrey", "white"), title = "order", legend = 2:5) writelabel("a") # Adams-Moulton stability.multistep(alpha = AdamsMoulton$alph[3,], beta = AdamsMoulton$beta[3,], xlim = c(-8, 1), ylim = c(-4, 4), fill = "black", main = "Adams-Moulton") stability.multistep(alpha = AdamsMoulton$alpha[4,], beta = AdamsMoulton$beta[4,], add = TRUE, fill = "darkgrey") stability.multistep(alpha = AdamsMoulton$alpha[5,], beta = AdamsMoulton$beta[5,], add = TRUE, fill = "lightgrey") legend("topleft", fill = c("black", "darkgrey", "lightgrey"), title = "order", legend = 3:5 ) writelabel("b") # 2nd-order BDF plot(0, type="n", xlim = c(-3, 12), ylim = c(-8, 8), main = "BDF order 2", xlab = "Re(z)", ylab = "Im(z)") rect(-100, -100, 100, 100, col = "lightgrey") box() stability.multistep (alpha = BDF$alpha[2,], beta = BDF$beta[2,], fill = "white", add = TRUE) writelabel("c") # 4th-order BDF

8 8 stability.multistep plot(0, type="n", xlim=c(-3, 12), ylim = c(-8, 8), main = "BDF order 4", xlab = "Re(z)", ylab = "Im(z)") rect(-100, -100, 100, 100, col = "lightgrey") box() stability.multistep (alpha = BDF$alpha[4,], beta = BDF$beta[4,], fill = "white", add = TRUE) writelabel("d") ## Stability regions for runge-kutta methods # Drawing the stability regions - brute force # stability function for explicit runge-kutta s rkstabfunc <- function (z, order = 1) { h <- 1 ss <- 1 for (p in 1: order) ss <- ss + (h*z)^p / factorial(p) return (abs(ss) <= 1) } # regions for stability orders 5 to 1 - rather crude Rez <- seq(-5, 1, by = 0.1) Imz <- seq(-3, 3, by = 0.1) stability.bruteforce(main = "Explicit RK", func = function(z) rkstabfunc(z, order = 5), Rez = Rez, Imz = Imz, fill = grey(0.95) ) stability.bruteforce(add = TRUE, func = function(z) rkstabfunc(z, order = 4), Rez = Rez, Imz = Imz, fill = grey(0.75) ) stability.bruteforce(add = TRUE, func = function(z) rkstabfunc(z, order = 3), Rez = Rez, Imz = Imz, fill = grey(0.55) ) stability.bruteforce(add = TRUE, func = function(z) rkstabfunc(z, order = 2), Rez = Rez, Imz = Imz, fill = grey(0.35) ) stability.bruteforce(add = TRUE, func = function(z) rkstabfunc(z, order = 1), Rez = Rez, Imz = Imz, fill = grey(0.15) ) legend("topleft", legend = 5:1, title = "order", fill = grey(c(0.95, 0.75, 0.55, 0.35, 0.15))) # stability function for radau method stability.bruteforce(main = "Radau 5", Rez = seq(-5,15,by=0.1), Imz = seq(-10,10,by=0.12),

9 stability.multistep 9 func = function(z) return(abs((1 + 2*z/5 + z^2/20) / (1-3*z/5 + 3*z^2/20 - z^3/60)) <= 1), col = grey(0.8) )

10 Index Topic datasets Coefficients, 3 Topic math stability.multistep, 6 Topic package diffeq-package, 2 Topic plot rkmethodplot, 4 AdamsBashforth (Coefficients), 3 AdamsMoulton (Coefficients), 3 BDF (Coefficients), 3 Coefficients, 2, 3 diffeq (diffeq-package), 2 diffeq-package, 2 rkmethodplot, 2, 4, 6 stability.bruteforce, 2, 5 stability.bruteforce (stability.multistep), 6 stability.multistep, 2, 3, 6 10